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            <ul>
<li><a class="reference internal" href="#">Outlier detection on a real data set</a><ul>
<li><a class="reference internal" href="#first-example">First example</a></li>
<li><a class="reference internal" href="#second-example">Second example</a></li>
</ul>
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  <div class="sphx-glr-download-link-note admonition note">
<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-applications-plot-outlier-detection-housing-py"><span class="std std-ref">here</span></a> to download the full example code or to run this example in your browser via Binder</p>
</div>
<div class="sphx-glr-example-title section" id="outlier-detection-on-a-real-data-set">
<span id="sphx-glr-auto-examples-applications-plot-outlier-detection-housing-py"></span><h1>Outlier detection on a real data set<a class="headerlink" href="#outlier-detection-on-a-real-data-set" title="Permalink to this headline">¶</a></h1>
<p>This example illustrates the need for robust covariance estimation
on a real data set. It is useful both for outlier detection and for
a better understanding of the data structure.</p>
<p>We selected two sets of two variables from the Boston housing data set
as an illustration of what kind of analysis can be done with several
outlier detection tools. For the purpose of visualization, we are working
with two-dimensional examples, but one should be aware that things are
not so trivial in high-dimension, as it will be pointed out.</p>
<p>In both examples below, the main result is that the empirical covariance
estimate, as a non-robust one, is highly influenced by the heterogeneous
structure of the observations. Although the robust covariance estimate is
able to focus on the main mode of the data distribution, it sticks to the
assumption that the data should be Gaussian distributed, yielding some biased
estimation of the data structure, but yet accurate to some extent.
The One-Class SVM does not assume any parametric form of the data distribution
and can therefore model the complex shape of the data much better.</p>
<div class="section" id="first-example">
<h2>First example<a class="headerlink" href="#first-example" title="Permalink to this headline">¶</a></h2>
<p>The first example illustrates how robust covariance estimation can help
concentrating on a relevant cluster when another one exists. Here, many
observations are confounded into one and break down the empirical covariance
estimation.
Of course, some screening tools would have pointed out the presence of two
clusters (Support Vector Machines, Gaussian Mixture Models, univariate
outlier detection, …). But had it been a high-dimensional example, none
of these could be applied that easily.</p>
</div>
<div class="section" id="second-example">
<h2>Second example<a class="headerlink" href="#second-example" title="Permalink to this headline">¶</a></h2>
<p>The second example shows the ability of the Minimum Covariance Determinant
robust estimator of covariance to concentrate on the main mode of the data
distribution: the location seems to be well estimated, although the covariance
is hard to estimate due to the banana-shaped distribution. Anyway, we can
get rid of some outlying observations.
The One-Class SVM is able to capture the real data structure, but the
difficulty is to adjust its kernel bandwidth parameter so as to obtain
a good compromise between the shape of the data scatter matrix and the
risk of over-fitting the data.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>

<span class="c1"># Author: Virgile Fritsch &lt;virgile.fritsch@inria.fr&gt;</span>
<span class="c1"># License: BSD 3 clause</span>

<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">sklearn.covariance</span> <span class="kn">import</span> <span class="n">EllipticEnvelope</span>
<span class="kn">from</span> <span class="nn">sklearn.svm</span> <span class="kn">import</span> <span class="n">OneClassSVM</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">matplotlib.font_manager</span>
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_boston</span>

<span class="c1"># Get data</span>
<span class="n">X1</span> <span class="o">=</span> <span class="n">load_boston</span><span class="p">()[</span><span class="s1">&#39;data&#39;</span><span class="p">][:,</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span> <span class="mi">10</span><span class="p">]]</span>  <span class="c1"># two clusters</span>
<span class="n">X2</span> <span class="o">=</span> <span class="n">load_boston</span><span class="p">()[</span><span class="s1">&#39;data&#39;</span><span class="p">][:,</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">12</span><span class="p">]]</span>  <span class="c1"># &quot;banana&quot;-shaped</span>

<span class="c1"># Define &quot;classifiers&quot; to be used</span>
<span class="n">classifiers</span> <span class="o">=</span> <span class="p">{</span>
    <span class="s2">&quot;Empirical Covariance&quot;</span><span class="p">:</span> <span class="n">EllipticEnvelope</span><span class="p">(</span><span class="n">support_fraction</span><span class="o">=</span><span class="mf">1.</span><span class="p">,</span>
                                             <span class="n">contamination</span><span class="o">=</span><span class="mf">0.261</span><span class="p">),</span>
    <span class="s2">&quot;Robust Covariance (Minimum Covariance Determinant)&quot;</span><span class="p">:</span>
    <span class="n">EllipticEnvelope</span><span class="p">(</span><span class="n">contamination</span><span class="o">=</span><span class="mf">0.261</span><span class="p">),</span>
    <span class="s2">&quot;OCSVM&quot;</span><span class="p">:</span> <span class="n">OneClassSVM</span><span class="p">(</span><span class="n">nu</span><span class="o">=</span><span class="mf">0.261</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">0.05</span><span class="p">)}</span>
<span class="n">colors</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;m&#39;</span><span class="p">,</span> <span class="s1">&#39;g&#39;</span><span class="p">,</span> <span class="s1">&#39;b&#39;</span><span class="p">]</span>
<span class="n">legend1</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">legend2</span> <span class="o">=</span> <span class="p">{}</span>

<span class="c1"># Learn a frontier for outlier detection with several classifiers</span>
<span class="n">xx1</span><span class="p">,</span> <span class="n">yy1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span> <span class="mi">28</span><span class="p">,</span> <span class="mi">500</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">40</span><span class="p">,</span> <span class="mi">500</span><span class="p">))</span>
<span class="n">xx2</span><span class="p">,</span> <span class="n">yy2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">500</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">45</span><span class="p">,</span> <span class="mi">500</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">clf_name</span><span class="p">,</span> <span class="n">clf</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">classifiers</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
    <span class="n">clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X1</span><span class="p">)</span>
    <span class="n">Z1</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">decision_function</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">xx1</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">yy1</span><span class="o">.</span><span class="n">ravel</span><span class="p">()])</span>
    <span class="n">Z1</span> <span class="o">=</span> <span class="n">Z1</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">xx1</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
    <span class="n">legend1</span><span class="p">[</span><span class="n">clf_name</span><span class="p">]</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span>
        <span class="n">xx1</span><span class="p">,</span> <span class="n">yy1</span><span class="p">,</span> <span class="n">Z1</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">linewidths</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">colors</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
    <span class="n">clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X2</span><span class="p">)</span>
    <span class="n">Z2</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">decision_function</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">xx2</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">yy2</span><span class="o">.</span><span class="n">ravel</span><span class="p">()])</span>
    <span class="n">Z2</span> <span class="o">=</span> <span class="n">Z2</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">xx2</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
    <span class="n">legend2</span><span class="p">[</span><span class="n">clf_name</span><span class="p">]</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span>
        <span class="n">xx2</span><span class="p">,</span> <span class="n">yy2</span><span class="p">,</span> <span class="n">Z2</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">linewidths</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">colors</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>

<span class="n">legend1_values_list</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">legend1</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
<span class="n">legend1_keys_list</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">legend1</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>

<span class="c1"># Plot the results (= shape of the data points cloud)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>  <span class="c1"># two clusters</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Outlier detection on a real data set (boston housing)&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X1</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X1</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">)</span>
<span class="n">bbox_args</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">boxstyle</span><span class="o">=</span><span class="s2">&quot;round&quot;</span><span class="p">,</span> <span class="n">fc</span><span class="o">=</span><span class="s2">&quot;0.8&quot;</span><span class="p">)</span>
<span class="n">arrow_args</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">arrowstyle</span><span class="o">=</span><span class="s2">&quot;-&gt;&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">annotate</span><span class="p">(</span><span class="s2">&quot;several confounded points&quot;</span><span class="p">,</span> <span class="n">xy</span><span class="o">=</span><span class="p">(</span><span class="mi">24</span><span class="p">,</span> <span class="mi">19</span><span class="p">),</span>
             <span class="n">xycoords</span><span class="o">=</span><span class="s2">&quot;data&quot;</span><span class="p">,</span> <span class="n">textcoords</span><span class="o">=</span><span class="s2">&quot;data&quot;</span><span class="p">,</span>
             <span class="n">xytext</span><span class="o">=</span><span class="p">(</span><span class="mi">13</span><span class="p">,</span> <span class="mi">10</span><span class="p">),</span> <span class="n">bbox</span><span class="o">=</span><span class="n">bbox_args</span><span class="p">,</span> <span class="n">arrowprops</span><span class="o">=</span><span class="n">arrow_args</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">((</span><span class="n">xx1</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">xx1</span><span class="o">.</span><span class="n">max</span><span class="p">()))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">((</span><span class="n">yy1</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">yy1</span><span class="o">.</span><span class="n">max</span><span class="p">()))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">((</span><span class="n">legend1_values_list</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">collections</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
            <span class="n">legend1_values_list</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">collections</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
            <span class="n">legend1_values_list</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">collections</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
           <span class="p">(</span><span class="n">legend1_keys_list</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">legend1_keys_list</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">legend1_keys_list</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span>
           <span class="n">loc</span><span class="o">=</span><span class="s2">&quot;upper center&quot;</span><span class="p">,</span>
           <span class="n">prop</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">font_manager</span><span class="o">.</span><span class="n">FontProperties</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">12</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;accessibility to radial highways&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;pupil-teacher ratio by town&quot;</span><span class="p">)</span>

<span class="n">legend2_values_list</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">legend2</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
<span class="n">legend2_keys_list</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">legend2</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>

<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>  <span class="c1"># &quot;banana&quot; shape</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Outlier detection on a real data set (boston housing)&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X2</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X2</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">((</span><span class="n">xx2</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">xx2</span><span class="o">.</span><span class="n">max</span><span class="p">()))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">((</span><span class="n">yy2</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">yy2</span><span class="o">.</span><span class="n">max</span><span class="p">()))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">((</span><span class="n">legend2_values_list</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">collections</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
            <span class="n">legend2_values_list</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">collections</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
            <span class="n">legend2_values_list</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">collections</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
           <span class="p">(</span><span class="n">legend2_keys_list</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">legend2_keys_list</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">legend2_keys_list</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span>
           <span class="n">loc</span><span class="o">=</span><span class="s2">&quot;upper center&quot;</span><span class="p">,</span>
           <span class="n">prop</span><span class="o">=</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">font_manager</span><span class="o">.</span><span class="n">FontProperties</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">12</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;</span><span class="si">% lo</span><span class="s2">wer status of the population&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;average number of rooms per dwelling&quot;</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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